Subgroups — #11757762

Order 972.

133 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉⋊A₄	C₃	1	1	yes	maximal, characteristic
2	C₂²×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₃×A₄)⋊C₉	C₃	1	3	yes	maximal
4	C₉⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
5	(C₂²⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
6	#6899	—	4	1	no	maximal, Sylow
7	C₂×C₉²	—	3	1	no	—
8	C₃²×A₄	C₃²	1	1	yes	characteristic
9	C₂²×C₃×C₉	C₃²	1	3	yes	—
10	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
11	C₃×C₂²⋊C₉	C₃²	1	2	yes	—
12	C₃×C₂²⋊C₉	C₃²	1	6	yes	—
13	C₉⋊A₄	—	3	3	no	—
14	C₃×3₋¹⁺²	—	4	1	no	—
15	C₉⋊C₉	—	4	6	no	—
16	C₉⋊C₉	—	4	2	no	—
17	C₃²⋊C₉	—	4	3	no	—
18	C₉²	A₄	1	1	yes	characteristic, complemented
19	C₂×C₃×C₉	—	3	3	no	—
20	C₂×C₃×C₉	—	3	1	no	—
21	C₂²×C₃²	C₃³	1	1	yes	characteristic
22	C₃×A₄	—	3	1	no	—
23	C₃×A₄	—	3	3	no	—
24	C₂²×C₉	—	3	3	no	—
25	C₂²×C₉	3₋¹⁺²	1	3	yes	—
26	C₂²⋊C₉	—	3	2	no	—
27	C₂²⋊C₉	—	3	6	no	—
28	C₃³	—	4	1	no	—
29	3₋¹⁺²	—	12	3	no	—
30	C₃×C₉	—	4	6	no	—
31	C₃×C₉	—	4	2	no	—
32	C₃×C₉	C₃×A₄	1	3	yes	—
33	C₃×C₉	C₃×A₄	1	1	yes	characteristic
34	C₂×C₃²	—	3	1	no	—
35	C₂×C₉	—	3	9	no	—
36	C₂×C₉	—	3	3	no	—
37	C₂²×C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
38	C₂²×C₃	#737	1	3	yes	—
39	A₄	—	9	1	no	—
40	C₃²	—	12	1	no	—
41	C₃²	—	12	3	no	—
42	C₃²	C₃²×A₄	1	1	yes	characteristic
43	C₉	—	3	3	no	—
44	C₉	—	12	2	no	—
45	C₉	—	12	6	no	—
46	C₉	C₉⋊A₄	1	3	yes	—
47	C₂×C₃	—	3	1	no	—
48	C₂×C₃	—	3	3	no	—
49	C₂²	#6899	1	1	yes	characteristic, complemented, Sylow
50	C₃	—	36	1	no	—
51	C₃	C₃×C₉⋊A₄	1	1	yes	characteristic
52	C₃	#66436	1	3	yes	—
53	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting	C₂²×C₉²	C₃
prime radical	C₃²×A₄	C₃²
hypercenter	C₉²	A₄
hypocenter	C₂²	#6899
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₉²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×C₉²	C₃	semidirect
4. C₉⋊(C₂²⋊C₉)	C₃	semidirect
5. (C₂²⋊C₉)⋊C₉	C₃	semidirect
18. C₉²	A₄	semidirect
49. C₂²	#6899	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₉⋊A₄	3
C₂²×C₉²	3
(C₃×A₄)⋊C₉	3
C₉⋊(C₂²⋊C₉)	3
(C₂²⋊C₉)⋊C₉	3
#6899	4